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If the function f is differentiable on an interval, its derived function f′ is defined. If f′ is also differentiable, then the derived function of this, denoted by f″, is the second derived function of f; its value at x, denoted by f″(x), or d2f/dx2, is the second derivative of f at x. (The term ‘second derivative’ may be used loosely also for the second derived function f″.)

Similarly, if f″ is differentiable, then f‴(x) or d3f/dx3, the third derivative of f at x, can be formed, and so on. The n-th derivative of f at x is denoted by f(n)(x) or dnf/dxn. The n-th derivatives, for n≥2, are called the higher derivatives of f. When y=f(x), the higher derivatives may be denoted by d2y/dx2,…, dny/dxn or y″, y‴,…, y(n). If, with a different notation, x is a function of t and the derivative dx/dt is denoted by ẋ the second derivative d2x/dt2 is denoted by ẍ